Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations

Abstract

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.

Existence Results for Block Matrix Operator of Fractional Orders in Banach Algebras

“…In this paper, we prove an existence theorem of solutions for a coupled system of quadratic integral equations of fractional order in Banach algebras, by a direct application of a block operator fixed point theorem [4]. This coupled system includes many key coupled systems of integral and differential equations that arise in nonlinear analysis and their applications.…”

“…where T 1 , T 2 , T 2 , T 3 , T 4 are nonlinear operators defined on Banach algebras. This kind of operators is studied by many researchers [4][5][6].…”

“…Kaddachi and et al [4] concentrated on answering the question: Under which conditions on its entries does the 2 × 2 operator matrix (Equation (1)) acting on a product of Banach algebras has a fixed point? In [4], some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras are studied, where the entries are nonlinear operators. Furthermore, the obtained results are applied to a coupled system of nonlinear equations.…”

Generalized form of fixed point theorems in Banach algebras under weak topology with an application

“…In [20,22], the authors have established some fixed point results for the block operator matrix (2), where the inputs are nonlinear mappings based on the convexity of the bounded domain, on the well-known Schauder's fixed point theorem, and also on the properties of the inputs (cf. completely continuous [22,25], weakly sequentially continuous [20], etc, . .…”

Existence of solutions for a system of Chandrasekhar’s equations in Banach algebras underweak topology

“…Recently, H. H. G. Hashem in [16] used some results of [19] to study the existence of solution for a system of quadratic integral equations of Chandrasekhar type by applying fixed point theorem for the block operator matrix (2) defined on a nonempty bounded closed convex subsets of Banach algebras where the entries are nonlinear operators.…”